Rigify moving and posing, origin doesn't seem to move
An armature moves the mesh data of an object, not the object itself. If you make the armature the parent of the mesh object, the mesh object will move with the armature when you move it in object mode. You will also notice that adjusting the armature in pose mode will leave the origin of the armature at the start position.

The modifiers have some display options that can be turned on/off. Some modifiers only have two some three and some four, the armature modifier has all four. Enabling all four will prevent the mesh snapping back to it's original position in edit mode.